A domestic violence case and voter registration irregularities involving Donald Trump’s new campaign chief executive brought fresh scrutiny on Friday to how well Trump vets his most senior employees and advisers — another distraction from the themes the GOP nominee wants to emphasize less than 11 weeks from the election.

Records show Stephen K. Bannon changed his voter registration address in Florida this week as reporters were preparing a story about how he was registered at an address where he did not live. A spokeswoman provided a statement from someone who said Bannon had lived there but did not respond to a que...